Some Extensions of the Mehler Formula

Abstract

Abstract.By using certain operational techniques, the authors prove an elegant unification of several extensions of the well-known Mehler formula for Hermite polynomials, given recently by L. Carlitz ([1], [2]). It is also shown how rapidly a number of Carlitz's formulas would follow from these considerations. The last section discusses a generalization involving the product of several Hermite polynomials of different arguments.

Hollow Gaussian Schell-model beam and its propagation

“…is the parabolic cylinder function, and ) (x H n is the Hermite polynomials. With the help of the extension form of Mehler's formula [30]: . Equation (12) is the general propagation formula for HGSMBs passing through the paraxial ABCD optical system.…”

“…is the parabolic cylinder function, and ) (x H n is the Hermite polynomials. With the help of the extension form of Mehler's formula [30]:…”